**12.215 Homework #3 Due
Wednesday December 2, 2009**

Statistics and Estimation

**Question 1:** Using the data from problem set
2, estimate using your own estimator (not polyfit)
the coefficients of the quadratic polynomial that best fit the data in a least
squares sense.

(a) Write the equation for the polynomial. What are the observations and unknown parameters in the
polynomial (5-points).

(b) Write the above set of equations in matrix form (5-points).

(c) Form the least squares estimator and solve for the coefficients.
You may use the Matlab matrix inversion routine inv
and/or a calculator matrix inversion to solve the system of equations (10
points).

**Question 2:** Estimate the standards deviation
of the errors in the measurements using the differences between the observed
values and the polynomial fit. (20-points)

**Question 3:** What
is the probability that the 12^{th} measurement (time 16hr 18m 21s Measurement 78^{o} 37.2Õ) differs from
the polynomial fit due to random error assuming that the noise in the
measurements is Gaussian distributed and the data standard deviation computed
in question 2 (20-points).

**Question 4**: Estimate the standard deviation
of the peak in the polynomial (i.e., its maximum value and the time at which
the maximum occurs) based on the least squares estimate in Question 1. From these results infer how well the
latitude and longitude were determined (20-points).

**Question 5**: Using the non-linear model for
the observed double elevation angle as a function of the site latitude and
longitude (and the Sun's declination and Greenwich hour angle which you can
assume are known), rigorously estimate the latitude and longitude of
campus. Approach the problem with
the following steps:

(a) Write the equation for the double-elevation as a function of
latitude and longitude (5-pts)

(b) Find the partial derivative of the double angle with respect to
latitude and longitude (10-pts)

(c) Find the differences between the measured double elevation angles
and the values predicted from an a priori estimate of the latitude and longitude
of 42.0 and 71.0 (5-pts)

(d) Using the partial derivatives from (b) and the prefit
residuals from (c), form the least squares solution for the estimates of the
adjustments to the latitude and longitude. Since this is a non-linear problem, iterate your solution
until the adjustments are small compared to the standard deviations of the
estimates i.e., compute new residuals from the non-linear model with the new
estimates of the latitude and longitude.
Use the estimates if the standard deviations of sextant measurements
from Q2 above. (10-pts)

(e) Give the final estimates of the latitude and longitude and their
standard deviations and correlation between the estimates. (10-pts)